Estudo de um controlador robusto via D-estabilidade através das inequações matriciais lineares aplicado a um quadcopter

Abstract

In this work, the robust controller analysis using the Lyapunov theory that addresses the problem of D-stability applied to a quadcopter through Linear Matrix Inequalities (LMIs) is presented. Several models of quadcopters have been developed for the most varied functions, like filming, taking photos, sending food and medicine to inaccessible places. The system's stability is essential to allow this equipment to perform that kind of functions. The analysis by the LMIs aims to guarantee the stability for the system allowing it to control the establishment time, overshoot and damped natural frequency of the movements yaw, pitch, roll and height of the equipment through the positions (x, y and z) and angulation (φ, θ and ψ) for each propeller. A mathematical modeling was developed for the quadcopter and a non-linear system was studied and linearized further. The D-stability problem aimed to check whether all poles of the system were contained in certain convex regions of the complex plane. Three LMIs regions were analyzed. From these three regions, it was possible to restrict the poles in the disc, cone and left half-plane region, which allowed the control of the damped frequency, the settling time and overshoot for the analyzed system. In order to examine the behavior of the system with respect to stability, both the linearized system and the non-linear system were submitted to the treatment of the D-stability problem. It was concluded that with the analyzes performed in the LMI regions, the overshoot, the damped natural frequency and the establishment time, presented the best responses to the system when the poles suffered the greatest restriction in the cone, disc and left half-plane regions, respectively.